First of all, you should always, if at all possible, create your logos as vectors to begin with. Converting a vector logo to a raster image in any size or format you need is no problem at all. Converting a raster logo to a vector is, depending on the size and format, not always easy. I loath having to vectorise low quality compressed image file logos, theres no reason it should even need to happen.
As for Image Trace—it's ok if you're tracing a high enough resolution image, but it's not meant for faithfully reproducing paths. If your logo is made up geometric shapes, straight lines etc. then you're better off recreating it by hand.
As for your actual question, you have a few options:
Create a compound path (Object → Compound Path → Make or cmd+8). This will leave all the path data intact, but you will have a single object that will be treated as such (for fills, strokes, effects etc.)
Pathfinder (Window → Pathfinder). The Unite command will merge shapes, this is (unless you Alt+click to create compound shapes) however a destructive process, so you lose all the distinct path data from your shapes. This normally is desirable on an output/deliverable file, but not in a working document.
You can read more about the different ways of combining shapes in Illustrator here: